e expansion in cold atoms
Download
Skip this Video
Download Presentation
e expansion in cold atoms

Loading in 2 Seconds...

play fullscreen
1 / 26

e expansion in cold atoms - PowerPoint PPT Presentation


  • 145 Views
  • Uploaded on

e expansion in cold atoms. Yusuke Nishida (Univ. of Tokyo & INT) in collaboration with D. T. Son (INT). Ref: Phys. Rev. Lett. 97, 050403 (2006). BCS-BEC crossover and unitarity limit Formulation of e (=4-d) expansion LO & NLO results Summary and outlook.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' e expansion in cold atoms' - amelia


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
e expansion in cold atoms

e expansion in cold atoms

Yusuke Nishida (Univ. of Tokyo & INT)

in collaboration with D.T. Son (INT)

Ref: Phys. Rev. Lett. 97, 050403 (2006)

  • BCS-BEC crossover and unitarity limit
  • Formulation of e(=4-d) expansion
  • LO & NLO results
  • Summary and outlook

21COE WS “Strongly correlated many-body systems” 19/Jan/07

interacting fermion systems
Interacting Fermion systems

Attraction Superconductivity/Superfluidity

    • Metallic superconductivity (electrons)
  • Kamerlingh Onnes (1911), Tc = ~9.2 K
    • Liquid 3He
  • Lee, Osheroff, Richardson (1972), Tc = 1~2.6 mK
    • High-Tc superconductivity (electrons or holes)
  • Bednorz and Müller (1986), Tc = ~160 K
    • Atomic gases (40K, 6Li)
  • Regal, Greiner, Jin (2003), Tc ~ 50 nK
    • Nuclear matter (neutron stars):?, Tc ~ 1MeV
    • Color superconductivity (quarks):??, Tc ~ 100MeV

BCS theory

(1957)

feshbach resonance
Feshbach resonance

C.A.Regal and D.S.Jin,

Phys.Rev.Lett. 90, (2003)

Attraction is arbitrarily tunable by magnetic field

S-wave scattering length :[0,]

Feshbach resonance

a (rBohr)

a>0

Bound state

formation

molecules

Strong coupling

|a|

a<0

No bound state

atoms

40K

Weak coupling

|a|0

bcs bec crossover
BCS-BEC crossover

Eagles (1969), Leggett (1980)

Nozières and Schmitt-Rink (1985)

Superfluid phase

-

+

0

BCS state of atoms

weak attraction:akF-0

BEC of molecules

weak repulsion:akF+0

Strong interaction : |akF|

  • Maximal S-wave cross section Unitarity limit
  • Threshold: Ebound = 1/(ma2)  0

Fermi gas in the strong coupling limit : akF=

Unitary Fermi gas

unitary fermi gas
Unitary Fermi gas

George Bertsch (1999),

“Many-Body X Challenge”

kF-1

r0

V0(a)

Atomic gas : r0=10Å << kF-1=100Å << |a|=1000Å

What are the ground state properties of

the many-body system composed of

spin-1/2 fermions interacting via a zero-range,

infinite scattering length contact interaction?

0r0 << kF-1<< a

kF is the only scale !

Energy per particle

x is independent of systems

cf.dilute neutron matter

|aNN|~18.5 fm >> r0~1.4 fm

universal parameter x
Universal parameterx
  • Strong coupling limit
  • Perturbation akF=
  • Difficulty for theory
  • No expansion parameter

Models

Simulations

Experiments

  • Mean field approx., Engelbrecht et al. (1996): x<0.59
  • Linked cluster expansion, Baker (1999): x=0.3~0.6
  • Galitskii approx., Heiselberg (2001): x=0.33
  • LOCV approx., Heiselberg (2004): x=0.46
  • Large d limit, Steel (’00)Schäfer et al. (’05): x=0.440.5
  • Carlson et al., Phys.Rev.Lett. (2003): x=0.44(1)
  • Astrakharchik et al., Phys.Rev.Lett. (2004): x=0.42(1)
  • Carlson and Reddy, Phys.Rev.Lett. (2005): x=0.42(1)

Duke(’03): 0.74(7), ENS(’03): 0.7(1), JILA(’03): 0.5(1),

Innsbruck(’04): 0.32(1), Duke(’05): 0.51(4), Rice(’06): 0.46(5).

Systematic expansion forx and other

observables (D,Tc,…) in terms ofe(=4-d)

This talk

2 body scattering around d 4
2-body scattering around d=4

iT

iT

2-component fermions

local 4-Fermi interaction :

2-body scattering at vacuum (m=0)

(p0,p) 

=

n

1

T-matrix at d=4-e(e<<1)

Coupling with boson

g = (8p2e)1/2/m

is SMALL !!!

ig

ig

=

iD(p0,p)

lagrangian for e expansion
Lagrangian for e expansion

Boson’s kinetic term is added,

and subtracted here.

  • Hubbard-Stratonovish trans. & Nambu-Gor’kov field :

=0 in dimensional regularization

Ground state at finite density is superfluid :

Expand with

  • Rewrite Lagrangian as a sum : L=L0+ L1+ L2
feynman rules 1
Feynman rules 1
  • L0 :
  • Free fermion quasiparticle  and boson 
  • L1 :

Small coupling “g” between  and 

(g~e1/2)

Chemical potential

insertions (m~e)

feynman rules 2
Feynman rules 2

k

p

p

=O(e)

+

p+k

k

p

p

p+k

=O(em)

+

  • L2 :

“Counter vertices” to

cancel 1/e singularities

in boson self-energies

1.

2.

O(e)

O(em)

power counting rule of e
Power counting rule ofe
  • Assume justified later
  • and consider to be O(1)
  • Draw Feynman diagrams using only L0 and L1
  • If there are subdiagrams of type
  • add vertices from L2 :
  • Its powers ofe will be Ng/2 + Nm
  • The only exception is = O(1) O(e)

or

or

Number of m insertions

Number of couplings “g ~e1/2”

thermodynamic functions at t 0
Thermodynamic functions at T=0
  • Universal equation of state
  • Effective potential and gap equation for 0

+ O(e2)

Veff (0,m) =

+

+

O(e)

O(1)

  • Universal numberxaround d=4

Systematic

expansion !

quasiparticle spectrum
Quasiparticle spectrum

p-k

k-p

-iS(p) =

+

p

p

p

p

k

k

  • O(e) fermion self-energy
  • Fermion dispersion relation : w(p)

Aroundminimum

Expansion over 4-d

Energy gap :

Location of min. :

0

extrapolation to d 3 from d 4 e
Extrapolation to d=3 from d=4-e
  • Keep LO & NLO resultsand extrapolate to e=1

NLO

corrections

are small

5 ~ 35 %

Good agreement with recent Monte Carlo data

J.Carlson and S.Reddy,

Phys.Rev.Lett.95, (2005)

cf. extrapolations from d=2+e

NLO are 100 %

matching of two expansions in x
Matching of two expansions in x
  • Borel transformation + Padé approximants

Expansion around 4d

x

♦=0.42

2d boundary condition

2d

  • Interpolated results to 3d

4d

d

summary
Summary
  • Systematic expansions over e=4-d
    • Unitary Fermi gas around d=4 becomes
    • weakly-interacting system of fermions & bosons
  • LO+NLO results onx, D, e0
    • NLO corrections around d=4 are small
    • Extrapolations to d=3 agree with recent MC data
  • Future problems
    • Large order behavior + NN…LO corrections
    • More understanding Precise determination

Picture of weakly-interacting fermionic &

bosonic quasiparticles for unitary Fermi gas may be a good starting point even at d=3

specialty of d 4 and 2
Specialty of d=4 and 2

Z.Nussinov and S.Nussinov, cond-mat/0410597

2-body wave function

Normalization at unitarity a

diverges at r0 for d4

Pair wave function is concentrated near its origin

Unitary Fermi gas for d4 is free “Bose” gas

At d2, any attractive potential leads to bound states

“a” corresponds to zero interaction

Unitary Fermi gas for d2 is free Fermi gas

specialty of d 4 and d 2
Specialty of d=4 and d=2

iT

2-component fermions

local 4-Fermi interaction :

2-body scattering in vacuum (m=0)

(p0,p) 

=

n

1

T-matrix at arbitrary spatial dimension d

“a”

Scattering amplitude has zeros at d=2,4,…

Non-interacting limits

t matrix around d 4 and 2
T-matrix around d=4 and 2

iT

iT

T-matrix at d=4-e(e<<1)

Small coupling b/w fermion-boson

g = (8p2e)1/2/m

ig

ig

=

iD(p0,p)

T-matrix at d=2+e(e<<1)

Small coupling b/w fermion-fermion

g = (2pe/m)1/2

ig2

=

unitary fermi gas at d 3
Unitary Fermi gas at d≠3

g

g

d=4

  • d4 : Weakly-interacting system of fermions & bosons, their coupling is g~(4-d)1/2

Strong coupling

Unitary regime

BEC

BCS

-

+

  • d2 : Weakly-interacting system of fermions, their coupling is g~(d-2)

d=2

Systematic expansions forx and other observables (D, Tc, …) in terms of “4-d” or “d-2”

expansion over e d 2
Expansion over e=d-2

Lagrangian

Power counting rule of

  • Assume justified later
  • and consider to be O(1)
  • Draw Feynman diagrams using only L0 and L1
  • If there are subdiagrams of type
  • add vertices from L2 :
  • Its powers ofe will be Ng/2
nnlo correction for x
NNLO correction for x

Arnold, Drut, and Son, cond-mat/0608477

  • O(e7/2) correction for x
  • Borel transformation + Padé approximants

x

  • Interpolation to 3d
  • NNLO 4d + NLO 2d
  • cf. NLO 4d + NLO 2d

NLO 4d

NLO 2d

d

NNLO 4d

critical temperature
Critical temperature
  • Gap equation at finite T

Veff = + + + minsertions

  • Critical temperature from d=4 and 2

NLO correctionis small ~4 %

Simulations :

  • Lee and Schäfer (’05): Tc/eF < 0.14
  • Burovski et al. (’06): Tc/eF = 0.152(7)
  • Akkineni et al. (’06): Tc/eF 0.25
  • Bulgac et al. (’05): Tc/eF = 0.23(2)
matching of two expansions t c
Matching of two expansions (Tc)

Tc/eF

4d

2d

d

  • Borel + Padé approx.
  • Interpolated results to 3d
e expansion in critical phenomena
eexpansion in critical phenomena

Critical exponents of O(n=1) 4 theory (e=4-d1)

  • Borel summation with conformal mapping
  • g=1.23550.0050 & =0.03600.0050
  • Boundary condition (exact value at d=2)
  • g=1.23800.0050 & =0.03650.0050

e expansion is

asymptotic series

but works well !

How about our case???

ad